Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of log-returns

The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)

vmr vhr vmrl pmr phr mmr mhr vmr_phr vhr_pmr
Min. : 0.868 0.849 0.801 0.904 0.878 0.988 0.977 0.979 0.967
1st Qu.: 1.044 1.039 1.013 1.042 1.068 1.013 1.013 1.021 1.011
Median : 1.097 1.099 1.085 1.084 1.128 1.085 1.113 1.102 1.094
Mean : 1.067 1.080 1.057 1.063 1.089 1.064 1.085 1.079 1.072
3rd Qu.: 1.136 1.160 1.128 1.107 1.182 1.101 1.128 1.121 1.107
Max. : 1.168 1.214 1.193 1.141 1.208 1.133 1.207 1.178 1.163

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.089 phr 1.182 phr 1.214 vhr
0.979 vmr_phr 1.044 vmr 1.113 mhr 1.085 mhr 1.160 vhr 1.208 phr
0.977 mhr 1.042 pmr 1.102 vmr_phr 1.080 vhr 1.136 vmr 1.207 mhr
0.967 vhr_pmr 1.039 vhr 1.099 vhr 1.079 vmr_phr 1.128 vmrl 1.193 vmrl
0.904 pmr 1.021 vmr_phr 1.097 vmr 1.072 vhr_pmr 1.128 mhr 1.178 vmr_phr
0.878 phr 1.013 vmrl 1.094 vhr_pmr 1.067 vmr 1.121 vmr_phr 1.168 vmr
0.868 vmr 1.013 mmr 1.085 vmrl 1.064 mmr 1.107 pmr 1.163 vhr_pmr
0.849 vhr 1.013 mhr 1.085 mmr 1.063 pmr 1.107 vhr_pmr 1.141 pmr
0.801 vmrl 1.011 vhr_pmr 1.084 pmr 1.057 vmrl 1.101 mmr 1.133 mmr

Correlations and covariance

Correlations

vmr vhr pmr phr
vmr 1.000 0.993 -0.197 -0.095
vhr 0.993 1.000 -0.119 -0.016
pmr -0.197 -0.119 1.000 0.957
phr -0.095 -0.016 0.957 1.000

Covariances

vmr vhr pmr phr
vmr 0.007 0.009 -0.001 -0.001
vhr 0.009 0.011 -0.001 0.000
pmr -0.001 -0.001 0.004 0.007
phr -0.001 0.000 0.007 0.011

Compare pension plans

Risk of loss

Risk of loss at least as big as x percent for a single period (year).
x values are row names.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 21.167 21.333 11.833 14.000 12.333 12.667 16.667 16.000
5 12.167 13.167 5.667 8.333 5.833 3.833 8.667 8.167
10 7.000 8.000 3.000 5.000 2.833 0.500 4.333 4.167
25 1.333 1.500 0.500 1.000 0.333 0.000 0.333 0.333
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Worst ranking for loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.333 vhr 13.167 vhr 8.000 vhr 1.500 vhr 0 vmr 0 vmr 0 vmr
21.167 vmr 12.167 vmr 7.000 vmr 1.333 vmr 0 vhr 0 vhr 0 vhr
16.667 vmr_phr 8.667 vmr_phr 5.000 phr 1.000 phr 0 pmr 0 pmr 0 pmr
16.000 vhr_pmr 8.333 phr 4.333 vmr_phr 0.500 pmr 0 phr 0 phr 0 phr
14.000 phr 8.167 vhr_pmr 4.167 vhr_pmr 0.333 mmr 0 mmr 0 mmr 0 mmr
12.667 mhr 5.833 mmr 3.000 pmr 0.333 vmr_phr 0 mhr 0 mhr 0 mhr
12.333 mmr 5.667 pmr 2.833 mmr 0.333 vhr_pmr 0 vmr_phr 0 vmr_phr 0 vmr_phr
11.833 pmr 3.833 mhr 0.500 mhr 0.000 mhr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Chance of min gains

Chance of gains of at least x percent for a single period (year).
x values are row names.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 78.833 78.667 88.167 86.000 87.667 87.333 83.333 84.000
5 63.833 66.667 71.667 76.000 71.667 70.167 69.333 69.000
10 40.833 50.167 32.500 59.667 35.500 46.000 47.167 43.833
25 0.000 0.000 0.000 0.000 0.000 0.833 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Best ranking for gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
88.167 pmr 76.000 phr 59.667 phr 0.833 mhr 0 vmr 0 vmr
87.667 mmr 71.667 pmr 50.167 vhr 0.000 vmr 0 vhr 0 vhr
87.333 mhr 71.667 mmr 47.167 vmr_phr 0.000 vhr 0 pmr 0 pmr
86.000 phr 70.167 mhr 46.000 mhr 0.000 pmr 0 phr 0 phr
84.000 vhr_pmr 69.333 vmr_phr 43.833 vhr_pmr 0.000 phr 0 mmr 0 mmr
83.333 vmr_phr 69.000 vhr_pmr 40.833 vmr 0.000 mmr 0 mhr 0 mhr
78.833 vmr 66.667 vhr 35.500 mmr 0.000 vmr_phr 0 vmr_phr 0 vmr_phr
78.667 vhr 63.833 vmr 32.500 pmr 0.000 vhr_pmr 0 vhr_pmr 0 vhr_pmr

MC risk percentiles

Risk of loss at least as big as row name in percent from first to last period.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 4.81 2.84 1.74 1.07 0.41 0.10 0.16 0.18
5 4.16 2.39 1.59 1.02 0.33 0.08 0.15 0.16
10 3.49 1.93 1.36 0.93 0.28 0.06 0.12 0.11
25 2.12 1.06 1.01 0.60 0.10 0.03 0.04 0.08
50 0.87 0.33 0.51 0.31 0.02 0.01 0.01 0.01
90 0.07 0.03 0.09 0.03 0.00 0.00 0.00 0.00
99 0.02 0.01 0.05 0.01 0.00 0.00 0.00 0.00

1e6 simulation paths of mhr:

0 5 10 25 50 90 99
prob_pct 0.118 0.095 0.076 0.036 0.008 0 0

Worst ranking for MC loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.81 vmr 4.16 vmr 3.49 vmr 2.12 vmr 0.87 vmr 0.09 pmr 0.05 pmr
2.84 vhr 2.39 vhr 1.93 vhr 1.06 vhr 0.51 pmr 0.07 vmr 0.02 vmr
1.74 pmr 1.59 pmr 1.36 pmr 1.01 pmr 0.33 vhr 0.03 vhr 0.01 vhr
1.07 phr 1.02 phr 0.93 phr 0.60 phr 0.31 phr 0.03 phr 0.01 phr
0.41 mmr 0.33 mmr 0.28 mmr 0.10 mmr 0.02 mmr 0.00 mmr 0.00 mmr
0.18 vhr_pmr 0.16 vhr_pmr 0.12 vmr_phr 0.08 vhr_pmr 0.01 mhr 0.00 mhr 0.00 mhr
0.16 vmr_phr 0.15 vmr_phr 0.11 vhr_pmr 0.04 vmr_phr 0.01 vmr_phr 0.00 vmr_phr 0.00 vmr_phr
0.10 mhr 0.08 mhr 0.06 mhr 0.03 mhr 0.01 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr

MC gains percentiles

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 95.19 97.16 98.26 98.93 99.59 99.90 99.84 99.82
5 94.52 96.84 98.03 98.84 99.49 99.89 99.77 99.78
10 93.73 96.43 97.84 98.68 99.37 99.88 99.75 99.74
25 91.25 94.88 97.12 98.20 98.85 99.79 99.50 99.53
50 85.80 91.67 95.25 97.31 97.56 99.47 98.99 98.90
100 72.20 83.01 88.55 94.67 89.93 97.65 96.24 94.23
200 39.42 60.89 59.51 85.27 48.53 86.42 80.34 66.41
300 16.21 39.24 22.32 70.63 11.13 62.93 52.20 30.58
400 5.17 23.88 4.42 54.36 1.26 37.79 25.70 9.69
500 1.51 12.81 0.54 38.37 0.09 18.77 9.84 2.54
1000 0.00 0.28 0.01 2.15 0.02 0.06 0.00 0.00

1e6 simulation paths of mhr:

0 5 10 25 50 100 200 300 400 500 1000
prob 99.882 99.854 99.824 99.686 99.301 97.513 86.912 65.992 41.486 21.693 0.086

Best ranking for MC gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
99.90 mhr 99.89 mhr 99.88 mhr 99.79 mhr 99.47 mhr 97.65 mhr
99.84 vmr_phr 99.78 vhr_pmr 99.75 vmr_phr 99.53 vhr_pmr 98.99 vmr_phr 96.24 vmr_phr
99.82 vhr_pmr 99.77 vmr_phr 99.74 vhr_pmr 99.50 vmr_phr 98.90 vhr_pmr 94.67 phr
99.59 mmr 99.49 mmr 99.37 mmr 98.85 mmr 97.56 mmr 94.23 vhr_pmr
98.93 phr 98.84 phr 98.68 phr 98.20 phr 97.31 phr 89.93 mmr
98.26 pmr 98.03 pmr 97.84 pmr 97.12 pmr 95.25 pmr 88.55 pmr
97.16 vhr 96.84 vhr 96.43 vhr 94.88 vhr 91.67 vhr 83.01 vhr
95.19 vmr 94.52 vmr 93.73 vmr 91.25 vmr 85.80 vmr 72.20 vmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
86.42 mhr 70.63 phr 54.36 phr 38.37 phr 2.15 phr
85.27 phr 62.93 mhr 37.79 mhr 18.77 mhr 0.28 vhr
80.34 vmr_phr 52.20 vmr_phr 25.70 vmr_phr 12.81 vhr 0.06 mhr
66.41 vhr_pmr 39.24 vhr 23.88 vhr 9.84 vmr_phr 0.02 mmr
60.89 vhr 30.58 vhr_pmr 9.69 vhr_pmr 2.54 vhr_pmr 0.01 pmr
59.51 pmr 22.32 pmr 5.17 vmr 1.51 vmr 0.00 vmr
48.53 mmr 16.21 vmr 4.42 pmr 0.54 pmr 0.00 vmr_phr
39.42 vmr 11.13 mmr 1.26 mmr 0.09 mmr 0.00 vhr_pmr

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.048 0.063 0.058 0.084 0.059 0.082 0.067 0.062
s 0.120 0.126 0.123 0.121 0.088 0.071 0.091 0.090
nu 3.304 4.390 2.265 3.185 2.773 89.863 4.660 3.892
xi 0.034 0.019 0.477 0.018 0.029 0.770 0.048 0.019
R^2 0.993 0.995 0.991 0.964 0.890 0.961 0.927 0.933

Fit statistics ranking

m ranking s ranking R^2 ranking
0.084 phr 0.071 mhr 0.995 vhr
0.082 mhr 0.088 mmr 0.993 vmr
0.067 vmr_phr 0.090 vhr_pmr 0.991 pmr
0.063 vhr 0.091 vmr_phr 0.964 phr
0.062 vhr_pmr 0.120 vmr 0.961 mhr
0.059 mmr 0.121 phr 0.933 vhr_pmr
0.058 pmr 0.123 pmr 0.927 vmr_phr
0.048 vmr 0.126 vhr 0.890 mmr

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 295.71 406.85 344.57 601.85 319.20 504.75 451.78 378.19
mc_s 135.11 211.26 114.07 273.01 102.36 173.81 152.58 121.44
mc_min 0.12 0.55 0.00 0.01 40.95 26.81 47.30 40.29
mc_max 1036.78 1504.07 1308.32 1930.64 4106.75 1414.95 1125.82 1097.82
dao_pct 0.00 0.00 0.03 0.01 0.00 0.00 0.00 0.00
dai_pct 4.47 2.46 1.66 0.99 0.42 0.10 0.14 0.15

Ranking

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
601.85 phr 102.36 mmr 47.30 vmr_phr 4106.75 mmr 0.00 vmr 0.10 mhr
504.75 mhr 114.07 pmr 40.95 mmr 1930.64 phr 0.00 vhr 0.14 vmr_phr
451.78 vmr_phr 121.44 vhr_pmr 40.29 vhr_pmr 1504.07 vhr 0.00 mmr 0.15 vhr_pmr
406.85 vhr 135.11 vmr 26.81 mhr 1414.95 mhr 0.00 mhr 0.42 mmr
378.19 vhr_pmr 152.58 vmr_phr 0.55 vhr 1308.32 pmr 0.00 vmr_phr 0.99 phr
344.57 pmr 173.81 mhr 0.12 vmr 1125.82 vmr_phr 0.00 vhr_pmr 1.66 pmr
319.20 mmr 211.26 vhr 0.01 phr 1097.82 vhr_pmr 0.01 phr 2.46 vhr
295.71 vmr 273.01 phr 0.00 pmr 1036.78 vmr 0.03 pmr 4.47 vmr

Compare Gaussian and skewed t-distribution fits

Gaussian fits

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.064 0.077 0.061 0.085 0.062 0.081 0.076 0.069
s 0.081 0.099 0.063 0.101 0.048 0.070 0.062 0.060

Gaussian QQ plots

Gaussian vs skewed t

Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
P_norm(X_min) 0.571 0.758 0.511 1.676 5.971 6.842 5.945 4.228
P_norm(X_max) 13.230 11.876 12.922 15.359 9.628 6.429 7.796 8.592
P_t(X_min) 5.377 5.080 3.489 4.315 10.570 8.015 13.008 10.520
P_t(X_max) 0.118 0.156 2.825 0.188 0.488 5.141 0.229 0.175

Average number of years between min or max events (respectively):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
norm: avg yrs btw min 175.248 131.911 195.568 59.669 16.748 14.616 16.820 23.650
norm: avg yrs btw max 7.559 8.420 7.739 6.511 10.386 15.556 12.827 11.639
t: avg yrs btw min 18.596 19.687 28.663 23.173 9.461 12.476 7.688 9.506
t: avg yrs btw max 848.548 640.410 35.400 531.552 205.104 19.450 437.280 572.483

Lilliefors test

p-values for Lilliefors test.
Testing \(H_0\), that log-returns are Gaussian.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
p value 0.052 0.343 0.024 0.06 0.24 0.137 0.375 0.415

Wittgenstein’s Ruler

For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?

Conditional probabilities for smallest observed log-returns:

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \leq x_{\text{min}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.052 0.343 0.024 0.060 0.240 0.137 0.375 0.415
Prior prob 0.948 0.657 0.976 0.940 0.760 0.863 0.625 0.585
P[Gauss | Event] 0.661 0.088 0.960 0.754 0.839 0.917 0.653 0.603

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \geq x_{\text{max}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.052 0.343 0.024 0.060 0.240 0.137 0.375 0.415
Prior prob 0.948 0.657 0.976 0.940 0.760 0.863 0.625 0.585
P[Gauss | Event] 1.000 0.986 0.997 0.998 0.993 0.888 0.988 0.991

Velliv medium risk (vmr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.2294983 0.3373312

Objective function plots

Velliv high risk (vhr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.5074609 0.4255322

Objective function plots

PFA medium risk (pmr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.2936284 0.3062685

Objective function plots

PFA high risk (phr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.8379614 0.4397688

Objective function plots

Mix medium risk (mmr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.1948623 0.2654885

Objective function plots

Mix high risk (mhr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.6413478 0.3380133

Objective function plots

Mix vmr+phr (vm_ph), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.5363616 0.3304634

Objective function plots

Mix vhr+pmr (mh_pm), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.3625460 0.3050122

Objective function plots

Comments

(Ignoring mhr_a…)

mhr has some nice properties:

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”

Appendix

Many simulations of mc_mhr: num_paths = 1e6

1e6 paths:

Compare \(10^6\) and \(10^4\) paths for mhr:

mc_m mc_s mc_min mc_max dao_pct dai_pct
mc_mhr_1e6 505.90695 173.22176 21.09569 1734.83520 0.00000 0.07330
mc_mhr_1e4 504.75125 173.80504 26.81367 1414.94530 0.00000 0.10000
is_mhr_1e4 510.836 2331.167 205.398 232384.846 ibid. ibid.

Arithmetic vs geometric mean

Let \(m\) be the number of steps in each path and \(n\) be the number of paths. \(a\) is the initial capital. Use arithmetic mean for mean of all paths at time \(t\): \[\dfrac{a (e^{z_1} + e^{z_2} + \dots + e^{z_n})}{n}\] where \[z_i := x_{i, 1} + x_{i, 2} + \dots + x_{i, m}\] Use geometric mean for mean of all steps in a single path \(i\): \[a e^{\frac{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}{m}} = a \sqrt[m]{e^{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}}\]

So for Monte Carlo of returns after \(m\) periods, we

  • fit a skewed t-distribution to log-returns and use that distribution to simulate \(\{x_{i, j}\}_j^m\),
  • for each path \(i\), calculate \(100\cdot e^{z_i}\),
  • calculate the mean of \(\{z_i\}_i^n\):
    • \[\bar{z} = 100\dfrac{e^{z_1} + e^{z_2} + \dots + e^{z_n}}{n}\]

For Importance Sampling, we

  • model log-returns on a skewed t-distribution,
  • for each path \(i\), calculate \(100\cdot e^{z_i}\),
  • fit a skewed t-distribution to \(\{z_i\}_i^n\) and use it as our \(f\) density function from which we simulate \(\{h_i\}_i^n\),
    • In our case \(h\) and \(z\) are identical, because we have an idea for a distribution to simulate \(z\), but in general for IS \(h\) could be a function of \(z\).
  • calculate \(w* = \frac{f}{g^*}\), where \(g*\) is our proposal distribution, which minimizes the variance of \(h\cdot w\).
  • calculate the arithmetic mean of \(\{h_i w_i^{*}\}_i^n\):
    • \[100 \dfrac{e^{h_1 w_1^{*}} + e^{h_2 w_2^{*}} + \dots + e^{h_n w_n^{*}}}{n}\]

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): 0.08883305 
## s(data_x): 0.4589474 
## m(data_y): 9.292777 
## s(data_y): 2.750134 
## 
## m(data_x + data_y): 4.690805 
## s(data_x + data_y): 1.346744

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
94.046 93.891 6.264 6.020
93.913 93.587 6.106 6.177
93.660 93.652 6.204 5.915
94.064 93.675 6.300 5.859
93.895 93.743 6.284 6.008
93.750 93.820 6.172 6.190
94.229 93.836 6.275 6.015
93.410 93.649 6.378 6.194
93.998 93.960 6.205 5.944
93.712 93.724 6.426 6.167 
##       m_a             m_b             s_a             s_b       
##  Min.   :93.41   Min.   :93.59   Min.   :6.106   Min.   :5.859  
##  1st Qu.:93.72   1st Qu.:93.66   1st Qu.:6.205   1st Qu.:5.960  
##  Median :93.90   Median :93.73   Median :6.269   Median :6.018  
##  Mean   :93.87   Mean   :93.75   Mean   :6.261   Mean   :6.049  
##  3rd Qu.:94.03   3rd Qu.:93.83   3rd Qu.:6.296   3rd Qu.:6.174  
##  Max.   :94.23   Max.   :93.96   Max.   :6.426   Max.   :6.194

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.06156   Min.   :0.04718  
##  1st Qu.:0.06885   1st Qu.:0.06157  
##  Median :0.07107   Median :0.07015  
##  Mean   :0.07218   Mean   :0.07064  
##  3rd Qu.:0.07693   3rd Qu.:0.08091  
##  Max.   :0.08582   Max.   :0.09479

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192